Robert W. Keener

Estimation of the Covariance Matrix of the Least Squares regression Coefficients when the Disturbance Covariance Matrix is of Unknown Form

This paper deals with the problem of estimating the covariance matrix of the least-squares regression coefficients under heteroskedasticity and/or autocorrelation of unknown form. We consider an estimator proposed by White [17] and give a relatively simple proof of its consistency. Our proof is based on more easily verifiable conditions than those of White. An alternative estimator with improved small sample properties is also presented.

A Generalized Parking Problem

Let F denote a distribution function which has a finite positive mean µ; let X 1 , X 2 ,... denote independent random variables with a common distribution function F; let S0, S1, S2,... denote the random walk S0 = 0 and S n = X 1 + ···+X n for n = 1, 2,... and let g denote a nonnegative finite convex function which attains its minimum at a unique b ≥ 0. The problem of minimizing E[g(S t )] with respect to stopping times t is considered. It is shown that there is an a a; and a is characterized in terms of a ladder height distribution.

Maximum Likelihood Regression of Rank-Censored Data

Abstract Linear regression is a common method for analyzing continuous, cardinal data, but it is inappropriate when the dependent variable is an ordinal ranking. The model proposed for analyzing these data sets is the general linear model u = Xβ + e, where the rank of the dependent variable u is observed instead of its value. A description is given for a numerical algorithm to evaluate the likelihood function that is efficient enough to permit maximum likelihood estimation of normalized regression coefficients. This algorithm can be modified to evaluate the cumulative distribution function of any multivariate normal random vector with nonsingular tridiagonal covariance matrix. Large sample properties of the maximum likelihood estimator are provided in the Appendix. Finite sample properties of the estimator are examined in a Monte Carlo experiment, and the exact finite sample distribution in one particular case is analyzed. The model is applied to voter preference data from a Louis Harris poll.

Tail probability approximations for U-statistics

Let Un be a second-degree, nondegenerate, zero mean U-statistic with a bounded kernel, scaled so that Un/[radical sign]n => N(0, [sigma]2). Large deviation approximations are developed for tail probabilities P(Un > x[radical sign]n) using a new explicit tilting procedure.

ASYMPTOTIC EXPANSIONS IN MULTIVARIATE RENEWAL THEORY

Let P be a probability measure , and R = [summation operator][infinity]n=0 P*n. the associated renewal measure. A two term asymptotic expansion for R is derived under moment and smoothness conditions. The smoothness conditions imposed allow P to be arithmetic is some coordinates and absolutely continuous in the other coordinates.

Fixed width interval estimation for the reciprocal drift of Brownian motion

Let b(t), 05 t< 03, denote Brownian motion with unknown positive drift p. The problem of setting a fixed width confidence interval for B= l/p is considered. The intervals studied are of the form (gC - h, e, + h), where { is a stopping time and & = (/b(c). Stopping times rh are derived so that these intervals have coverage probabilities converging to a set value y as h + 0. This convergence is uniform for p near 0. Asymptotic optimality of r/, is also addressed.

A nonlinear parking problem

The problem of minimizing Egc(Zt) is considered asymptotically, where Z0,Z1,… is a perturbed random walk and gc are convex functions which are minimized at values that approach ∞ as c ↓ 0. It is shown that a first passage time is asymptotically optimal, and the boundary for this time is characterized in terms of gc and the limiting distribution of the excess over the boundary. Applications to change point problems and power one tests are presented.

Asymptotic expansions for renewal measures in the plane

SummaryLet P be a distribution in the plane and define the renewal measure R=ΣP*nwhere * denotes convolution. The main results of this paper are three term asymptotic expansions for R far from the origin. As an application, expansions are obtained for distributions in linear boundary crossing problems.

Multi-armed bandits with simple arms

An exact solution to certain multi-armed bandit problems with independent and simple arms is presented. An arm is simple if the observations associated with the arm have one of two distributions conditional on the value of an unknown dichotomous parameter. This solution is obtained relating Gittins indices for the arms to ladder variables for associated random walks.

Probability and Measure

Much of the theory of statistical inference can be appreciated without a detailed understanding of probability or measure theory. This book does not treat these topics with rigor. But some basic knowledge of them is quite useful. Much of the literature in statistics uses measure theory and is inaccessible to anyone unfamiliar with the basic notation. Also, the notation of measure theory allows one to merge results for discrete and continuous random variables. In addition, the notation can handle interesting and important applications involving censoring or truncation in which a random variable of interest is neither discrete nor continuous. Finally, the language of measure theory is necessary for stating many results correctly. In the sequel, measure-theoretic details are generally downplayed or ignored in proofs, but the presentation is detailed enough that anyone with a good background in probability should be able to fill in any missing details.